I think I played this hand far too passively. Should I have just taken the initiative on the flop with my big draw? The difference in stack sizes between the opponents also made me confused as to the correct action

I'd most likely just Check/Raise the turn and might even play for the stacks. BB may easily have a lot of worse hands and maybe we need even to protect our hand. Another possibility would be also to Check/Raise the flop right there since we are planning to play for the stacks most likely.